11 entries
enumerative combinatorics probability theory
Any five distinct numbers contain three that rise in order or three that fall in order. Erdős and Szekeres computed the exact length that forces it, which turns a run into a measurement of the window rather than of the series.
Joan of Arc has one of the densest records of any medieval life, and her prosecutors produced it. Four properties get collapsed into the phrase well documented, and they trade off against each other rather than moving together.
probability distributions experimental design
A distributional estimate can be mathematically tidy while the design that generated its observations left whole groups outside it. Distribution parameters and experimental allocation are two halves of the same claim.
A function can be continued beyond an observed region only because it belongs to a class that constrains every extension. Complex analysis makes the constraint exact; a fitted curve should disclose when it does not have one.
mathematical logic computability
Finite examples can make a rule engine look complete even when no procedure can decide every case it will meet. Logic separates consistency, completeness and decidability; computability supplies the hard boundary.
Agreement on the text does not produce agreement on the rule for reading it. Textual canons and theology show that disputes often begin after the primary record has been fixed.
combinatorial designs graph algorithms
A shortest or cheapest route can look diversified because it has many hops while repeatedly relying on one untested pairing. Combinatorial designs supply the coverage constraint that graph optimization does not.
topology non euclidean geometry
Two objects can remain neighbours under a representation while the distances, angles and apparent densities are wrong. Topology supplies invariants; non-Euclidean geometry supplies the metric that the drawing hid.
The New Testament survives in about 25,000 manuscripts. Somewhere between three witnesses and several thousand, counting documents stops being the same activity as counting sources.
chronology genealogy graph algorithms
Three disciplines that will not take their own word for a result, and the three different things they each mean by independent.
Thirty generations back you should have a billion ancestors. There weren't a billion people. The arithmetic that fails there fails the same way in an exposure report.